( viii) Vector Product of Two Unit Vectors If a and b are unit vectors, then a = | a | = 1, b = | b | = 1 ∴ a * b = ab sin θ n =( sin theta;). n
( ix) Vector Product is not Commutative The two vector products a * b and b * a are equal in magnitude but opposite in direction.
i. e., b * a =- a * b ……..( i)
( x) The vector product of a vector a with itself is null vector, i. e., a * a = 0.
( xi) Distributive Law For any three vectors a, b, c a *( b + c) =( a * b) +( a * c)
( xii) Area of a Triangle and Parallelogram
( a) The vector area of a ΔABC is equal to 1 / 2 | AB * AC | or 1 / 2 | BC * BA | or 1 / 2 | CB *
CA |.
( b) The area of a ΔABC with vertices having PV’ s a, b, c respectively, is 1 / 2 | a * b + b * c + c
* a |.
( c) The points whose PV’ s are a, b, c are collinear, if and only if a * b + b * c + c * a
( d) The area of a parallelogram with adjacent sides a and b is | a * b |.
( e) The area of a Parallelogram with diagonals a and b is 1 / 2 | a * b |.
( f) The area of a quadrilateral ABCD is equal to 1 / 2 | AC * BD |.